Quillen's conjecture in rational homology

1 · The essentials

The manuscript studies finite mathematical groups through the smaller groups they contain. For each prime number, it selects smaller groups of a particular kind and arranges them according to containment. The question is whether the resulting shape can be continuously collapsed to a single point. Under a specific condition on the original group, the manuscript claims that this shape has a nonzero mathematical signal, measured using rational numbers, which rules out such a collapse. The condition is that the largest normal subgroup whose size is a power of the chosen prime consists only of the identity. The claimed result covers every finite group and every prime subject to that condition. It has not been formalized in Lean or formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a map of nested storage boxes, recording which boxes fit inside which others rather than their physical dimensions. The manuscript examines a shape built from such a containment map and claims that, under its stated condition, the shape retains a feature that prevents it from collapsing to one point.

Why it matters

The source presents the result as establishing the rational-homology form of Quillen's conjecture, which it describes as a stronger form of the conjecture. Its significance is the claimed uniform scope: every finite group and every prime satisfying the stated hypothesis, rather than a restricted class. The nonzero homology conclusion also gives the stated consequence that the associated shape is not contractible.

2 · Going further

Let $G$ be a finite group and $p$ a prime. Write $O_{p} (G)$ for the largest normal $p$ -subgroup of $G$ , and consider the poset of nontrivial elementary abelian $p$ -subgroups, ordered by inclusion. The precise claim is that if $O_{p} (G) = 1$ , this poset has nonzero augmented reduced homology with rational coefficients. Consequently, it is not contractible.

The repository summary and the main abstract both identify this as the stronger rational-homology form of Quillen's conjecture. Here the homology statement is the central conclusion, and noncontractibility is its stated consequence. The word “augmented” is part of the supplied formulation and is retained in stating the result.

The abstract supplies the hypothesis and conclusion but does not describe the argument, intermediate constructions, or reductions. Thus the only logical route available from the supplied text is the claimed nonvanishing of rational homology followed by noncontractibility; no method for obtaining that nonvanishing is given. Only the main manuscript, “Rational homology and Quillen's conjecture,” is supplied, so no relationship to companion manuscripts can be described.

Useful notions

Finite groups and subgroups
Normal subgroups and prime-power orders
Elementary abelian groups
Partially ordered sets under inclusion
Reduced homology and contractibility

What was known before / what is new

Before

The source identifies Quillen's conjecture as the problem being addressed and distinguishes its rational-homology form as stronger. It does not describe earlier partial results, the history of the conjecture, or previously available methods.

New

The manuscript claims nonzero augmented reduced rational homology for the poset of nontrivial elementary abelian $p$ -subgroups whenever $O_{p} (G) = 1$ . It states this for every finite group and every prime, and presents it as establishing the stronger rational-homology form of Quillen's conjecture.

Limits and assumptions

The claim assumes that $G$ is finite, $p$ is prime, and $O_{p} (G) = 1$ ; the supplied statement gives no conclusion when that last condition fails. Its homology conclusion uses rational coefficients. The abstract does not provide an argument to assess, and the main result is not formalized in Lean and has not been formally verified. The supplied text names no remaining open questions, so none can be identified from it.

Mini glossary

Largest normal p-subgroup: The subgroup denoted $O_{p} (G)$ is the largest subgroup of $G$ that is invariant under conjugation and has order a power of $p$ .
Elementary abelian p-subgroup: An elementary abelian $p$ -subgroup is a commutative subgroup in which every nonidentity element has order $p$ .
Poset: A poset is a set with a partial ordering, here the ordering of subgroups by inclusion.
Rational homology: Rational homology is an algebraic description of features of a shape using rational coefficients.
Contractible: A shape is contractible if it can be continuously deformed to a single point.

Keywords : Quillen's conjecture · Rational homology · Finite groups

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Rational homology and Quillen's conjecture

24/09/2026

To be verified

We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup $O_{p} (G)$ , then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.

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